Meulensteen's departure from the Premier League champions had been expected following Sir Alex Ferguson's retirement at the end of last season.
The Dutchman has become the third senior member of staff to leave Carrington following the exits of assistant manager Mike Phelan and goalkeeping coach Eric Steele in May, and he is now excited for a new challenge in Russia.
"I'm looking forward to it and am glad that things with Manchester United have been completed," he told Algemeen Dagblad.
Meulensteen had been in charge of first-team training as Sir Alex delegated more and more of his daily routine towards the end of his Old Trafford tenure.
But while reports claim the 49-year-old was offered a more senior role at United next season, he has suggested that incoming manager David Moyes' hands-on approach would lessen his own importance.
"My responsibilities at the club had grown in recent years," he continued. "Moyes is a coach who is still on the training field."